Problem: Simplify; express your answer in exponential form. Assume $y\neq 0, r\neq 0$. $\dfrac{{(y^{5}r^{-4})^{-5}}}{{(y^{5}r^{2})^{-2}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(y^{5}r^{-4})^{-5} = (y^{5})^{-5}(r^{-4})^{-5}}$ On the left, we have ${y^{5}}$ to the exponent ${-5}$ . Now ${5 \times -5 = -25}$ , so ${(y^{5})^{-5} = y^{-25}}$ Apply the ideas above to simplify the equation. $\dfrac{{(y^{5}r^{-4})^{-5}}}{{(y^{5}r^{2})^{-2}}} = \dfrac{{y^{-25}r^{20}}}{{y^{-10}r^{-4}}}$ Break up the equation by variable and simplify. $\dfrac{{y^{-25}r^{20}}}{{y^{-10}r^{-4}}} = \dfrac{{y^{-25}}}{{y^{-10}}} \cdot \dfrac{{r^{20}}}{{r^{-4}}} = y^{{-25} - {(-10)}} \cdot r^{{20} - {(-4)}} = y^{-15}r^{24}$